#27530907

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1885 ⋊ C₂
Sylow	C₅₃ ⋊ #32696

Subgroups

	subgroup	G/H
center	C₂	#20281942
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1885	C₂
socle	C₂×C₅₃	#2749
prime radical	#16811190	C₂²
hypercenter	#1885	D₅₃
hypocenter	C₅₃	#32696
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1885

552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 85 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(55,62)(58,73)(61,68)(67,72)(69,85)(71,76)(75,80)(77,83);(54,57)(55,61,62,68)(56,65,66,82)(58,71)(59,70)(60,74)(63,78,79,84)(64,81)(67,69,72,85)(73,76)(75,77)(80,83);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,56)(55,60)(57,69)(58,72)(59,66)(61,77)(62,64)(63,80)(65,71)(67,73)(68,83)(70,85)(74,78)(75,79)(76,82)(81,84);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55,58,63,59,62,73,79)(56,64,67,80,66,60,72,75)(57,68,71,84,70,61,76,78)(65,81,69,83,82,74,85,77)

Polycyclic presentation:

2^8,53|a^2=g,c^2=ef,g^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,i^a=i^52,c^b=cg,d^b=de,f^b=fh,g^b=gh,i^b=i^52,e^c=eh,f^c=fh,g^d=gh

Conjugacy classes & character table

866 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 316383);